
We note that the norm constraint here has dimension \(c\)  and \(\mathbf {w}_1\)  has dimension \(sd\) , so by [1]}, [2]} this strong duality result from Lemma REF  requires that \(m^* \le \min \lbrace sd, c\rbrace \) . This is equivalent to
\(p_{FN}^* &= \max _{\mathbf {V}_i} -\sum _{i=1}^n \mathcal {L}^*(\mathbf {V}_i, \mathbf {Y}_i) \nonumber \\\mathrm {s.t.}~& \Vert \sum _{i=1}^n \mathbf {V}_i^\top \mathrm {circ}({\mathbf {X}}_i)\Vert _2 \le \beta \) 
